Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 11Citation - Scopus: 12Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types(IOP Publishing Ltd., 2016) Aslan, İsmailDifferential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous. Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations, the majority of the results deal with polynomial types. Limited research has been reported regarding such equations of rational type. In this paper we present an adaptation of the (G′/G)-expansion method to solve nonlinear rational differential-difference equations. The procedure is demonstrated using two distinct equations. Our approach allows one to construct three types of exact traveling wave solutions (hyperbolic, trigonometric, and rational) by means of the simplified form of the auxiliary equation method with reduced parameters. Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.Article Citation - WoS: 1Citation - Scopus: 2Application of the Division Theorem To Nonlinear Physical Models for Constructing Traveling Waves(Politechnica University of Bucharest, 2013) Aslan, İsmailWe extend the so-called first integral method, which is based on the division theorem, to the Sharma-Tasso-Olver equation and the (2+1)-dimensional modified Boussinesq equation. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Traveling wave solutions are constructed through the established first integrals.Article Citation - WoS: 10Citation - Scopus: 11Some Exact and Explicit Solutions for Nonlinear Schrödinger Equations(Polish Academy of Sciences, 2013) Aslan, İsmailNonlinear models occur in many areas of applied physical sciences. This paper presents the first integral method to carry out the integration of Schrödinger-type equations in terms of traveling wave solutions. Through the established first integrals, exact traveling wave solutions are obtained under some parameter conditions.Article Citation - WoS: 23Citation - Scopus: 26Travelling Wave Solutions To Nonlinear Physical Models by Means of the First Integral Method(Springer Verlag, 2011) Aslan, İsmailThis paper presents the first integral method to carry out the integration of nonlinear partial differential equations in terms of travelling wave solutions. For illustration, three important equations of mathematical physics are analytically investigated. Through the established first integrals, exact solutions are successfully constructed for the equations considered. © Indian Academy of Sciences.Article Citation - WoS: 34Citation - Scopus: 34Exact and Explicit Solutions To Nonlinear Evolution Equations Using the Division Theorem(Elsevier Ltd., 2011) Aslan, İsmailIn this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson-Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner.Article Citation - WoS: 9Citation - Scopus: 10Application of the Exp-Function Method To Nonlinear Lattice Differential Equations for Multi-Wave and Rational Solutions(John Wiley and Sons Inc., 2011) Aslan, İsmailIn this paper, we extend the basic Exp-function method to nonlinear lattice differential equations for constructing multi-wave and rational solutions for the first time. We consider a differential-difference analogue of the Korteweg-de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant.Article Citation - WoS: 6Citation - Scopus: 6Some Exact Solutions for Toda Type Lattice Differential Equations Using the Improved (g'/g)-expansion Method(John Wiley and Sons Inc., 2012) Aslan, İsmailNonlinear lattice differential equations (also known as differential-difference equations) appear in many applications. They can be thought of as hybrid systems for the inclusion of both discrete and continuous variables. On the basis of an improved version of the basic (G′/G)- expansion method, we focus our attention towards some Toda type lattice differential systems for constructing further exact traveling wave solutions. Our method provides not only solitary and periodic wave profiles but also rational solutions with more arbitrary parameters. © 2012 John Wiley & Sons, Ltd.Article Citation - WoS: 32Citation - Scopus: 37Symbolic Computation and Construction of New Exact Traveling Wave Solutions To Fitzhugh-Nagumo and Klein-Gordon Equations(Walter de Gruyter GmbH, 2009) Öziş, Turgut; Aslan, İsmailWith the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.Article Citation - WoS: 77Citation - Scopus: 110Analytic Study on Two Nonlinear Evolution Equations by Using the (g'/g)-expansion Method(Elsevier Ltd., 2009) Aslan, İsmail; Öziş, TurgutThe validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.Article Citation - WoS: 16Citation - Scopus: 16Analytic Investigation of the (2 + 1)-Dimensional Schwarzian Korteweg–de Vries Equation for Traveling Wave Solutions(Elsevier Ltd., 2011) Aslan, İsmailBy means of the two distinct methods, the Exp-function method and the extended (G0/G)-expansion method, we successfully performed an analytic study on the (2 + 1)-dimensional Schwarzian Korteweg–de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.